"The agile mind is pleased to find what it was not looking for."Also just read an essay on mathematics and creativity -- A Mathematician's Lament -- by Paul Lockhart -- a damning critique of the typical teaching of mathematics -- devoid of the recognition of its inherent creativity. I want every teacher who teaches mathematics to read this and justify their current practices to me (says the indignant librarian).
"Wandering is the trick, and giving up on 'loss' or 'gain', and then agility of mind."
Dumb luck = luck of chance
Smart luck = craft added to accident, i.e, "a kind of responsive intelligence invoked by whatever happens"
Louis Pasteur quote: "chance favors the prepared mind", i.e., a mind prepared for what it isn't prepared for...
Chogyam Trungpa quote: "magic is the total appreciation of chance"
creation (absolute newness) vs. revelation (accident as a tool of revelation)
Absolute newness = "in a civilization as complex and shifting as ours has become, a readiness to let the mind change as contingency demands may be one prerequisite of a happy life."
Everything he says rings true to me because I had a teacher like Paul Lockhart from 7th grade onwards in my little town in Maine. Thank god for Wally Hayes and Ralph (Danny) Small. They made mathematics come alive -- and made us exercise our mental creativity every day in the name of mathematics. It was pure theater at times -- how Mr. Small would enthuse over a new proof he'd thought up the night before. We believed him when he said he had the quadratic formula framed over his bedstead. We never doubted that he spent his evenings reading mathematics books, enhanced by a bowl of potato chips and a glass of milk.
He never used labels for what he showed us -- he just showed us his thinking and encouraged us to show him ours. I went off to college/university believing I'd never had calculus, because that word had never come up. So I ended up repeating almost a whole year, not really knowing where Mr. Small had left off. But I definitely recognized that what my college professor had to offer was lesser stuff -- it was all just "cookbook" mathematics, whereas I had been trained to do the real thing -- proofs and analysis and an underlying understanding all along, no matter whether I knew what the outside world called it or not.
In reading Malcolm Gladwell's "Outliers" -- and the luck of Bill Gates to have access to a computer in junior high in Washington state in 1967 or whenever it was -- I can't help but realize that I also was lucky. It was 1972 when I was a freshman in high school that I got to program for the first time. We had a time-share set-up with some computer in Portland and all of us had to write a computer program to solve the quadratic formula. That meant creating an oiled punched-out tape that got fed into the remote reader and loaded into memory. So at 15 I began my relationship with computers (okay, nowhere near the 10,000 deliberate practice hours of a Gates, but..). Maybe it isn't so surprising that in 1980 I ended up in Boston at a software development company, despite my major in Russian Civilization. (I always used to say, languages are languages -- whether natural or mechanical.)
Anyway, here's Paul Lockhart in full force on how math should be considered in the curriculum:
"The first thing to understand is that mathematics is an art. The difference between math and the other arts, such as music and painting, is that our culture does not recognize it as such."Image credit: gadl via flickr
"A mathematician, like a painter or poet, is a maker of patterns. If his patterns are more permanet than theirs, it is because they are made with ideas." [G.H. Hardy]
"Mathematicians enjoy thinking about the simplest possible things, and the simplest possible things are imaginary."
"Mathematics is the art of explanation. If you deny students the opportunity to engage in this activity.... you deny them mathematics itself."
"Math is not about following directions, it's about making new directions."
"A piece of mathematics is like a poem, and we can ask if it satisfies our aesthetic criteria: Is this argument sound? Does it make sense? Is it simple and elegant? Does it get me closer to the heart of the matter?"
"Mathematics is the music of reason."
"The trouble is that math, like painting or poetry, is hard creative work. That makes it very difficult to teach. Mathematics is a slow, contemplative process. It takes time to produce a work of art, and it takes a skilled teacher to recognize one."
"Teaching is not about information. It's about having an honest intellectual relationshiop with your students."
"How ironic that people dismiss mathematics as the antithesis of creativity. They are missing out on an art form older than any book, more profound than any poem, and more abstract that any abstract."